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Birkhäuser Boston eBooks

Continuous-Time Regularized Path Integrals

John R. Klauder

Published Oct 28, 2010• 1 citation
Publisher

Abstract

Up to this point we have focused on temporal lattice regularizations in order to give meaning to the traditional configuration and phase space path integrals including coherent state phase space path integrals. In this chapter, we take up the question of introducing a continuous-time regularization procedure into a formal phase space path integral. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-0-8176-4791-9_8

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Up to this point we have focused on temporal lattice regularizations in order to give meaning to the traditional configuration and phase space path integrals including coherent state phase space path integrals. In this chapter, we take up the question of introducing a continuous-time regularization procedure into a formal phase space path integral. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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By introducing a suitable continuous-time regularization into a formal phase-space path integral it follows that the propagator is given by the limit of well-defined functional integrals involving standard stochastic processes and their associated probability measures. Such regularizations require pinning of both coordinate and momenta variables, and automatically lead to coherent-state representations. It is found that each standard independent increment process, involving a superposition of a Wiener and a Poisson process, is associated with a specific, generally non-Gaussian, fiducial vector with which the coherent states are defined.
Efforts to give an improved mathematical meaning to Feynman's path integral formulation of quantum mechanics started soon after its introduction and continue to this day. In the present paper, one common thread of development is followed over many years, with contributions made by various authors. The present version of this line of development involves a continuous-time regularization for a general phase space path integral and provides, in the author's opinion at least, perhaps the optimal formulation of the path integral.